Each morphism \({{Y}}_c \times {{Y}}_\mathbb {R}\rightarrow {{Y}}_c\)  in \(I\)  is a morphism between cofibrant objects in \({H}_\infty ^{p/i}\) .
Therefore, by [1]} it suffices to show that \(S_e\)  sends each morphism in \(I\)  to a weak equivalence in \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\) .
One way to see that this holds true is by observing that \(S_e\)  preserves finite products (it is a monoidal functor).
Therefore, we have that
\(S_e({{Y}}_c \times {{Y}}_\mathbb {R}\rightarrow {{Y}}_c)&= S_e \big ( {{Y}}_c \xrightarrow{} {{Y}}_c \big )\times S_e \big ( {{Y}}_\mathbb {R}\xrightarrow{} * \big )\\&= S_e (1_{{{Y}}_c}) \otimes S_e \big ( {{Y}}_\mathbb {R}\xrightarrow{} * \big )\,,\) 
